14.1 Properties of Numbers
Note
Associative Laws
Addition:
a
,
b
, and
c
are any numbers, then
(
a
+
b
)
+
c
=
a
+
(
b
+
c
)
.
Multiplication
a
,
b
, and
c
are any numbers, then
(
a
⋅
b
)
⋅
c
=
a
⋅
(
b
⋅
c
)
.
Note
Commutative Laws
Addition:
a
and
b
are any numbers, then
a
+
b
=
b
+
a
.
Multiplication
a
and
b
are any numbers, then
a
⋅
b
=
b
⋅
a
.
Note
Distributive Law
a
(
b
+
c
)
=
a
b
+
a
c
for any numbers
a
,
b
, and
c
.
Note
Properties of Equality
Addition:
a
=
b
and
c
is any number, then
a
+
c
=
b
+
c
.
Subtraction:
a
=
b
and
c
is any number, then
a
−
c
=
b
−
c
.
Multiplication
a
=
b
and
c
is any number, then
a
⋅
c
=
b
⋅
c
.
Division
a
=
b
and
c
is any nonzero number, then
a
c
=
b
c
.
Note
Fundamental Principle of Fractions
If
a
is any number, and
b
and
c
are nonzero numbers, then
a
⋅
c
b
⋅
c
=
a
b
.
Note
Laws of Exponents
a
m
⋅
a
n
=
a
m
+
n
a
m
a
n
=
a
m
−
n
b
l
a
n
k
1
(
n
<
m
)
a
m
a
n
=
1
a
n
−
m
b
l
a
n
k
(
n
>
m
)
(
a
m
)
n
=
a
m
+
n
(
a
b
)
n
=
a
n
b
n
(
a
b
)
n
=
a
n
b
n
Note
Product Rule for Radicals
If
a
and
b
are both nonnegative, then
a
b
=
a
b
.
Note
Quotient Rule for Radicals
If
a
≥
0
and
b
>
0
, then
a
b
=
a
b
.
Note
Properties of Absolute Value
|
a
+
b
|
≤
|
a
|
+
|
b
|
Triangle inequality
|
a
b
|
=
|
a
|
|
b
|
Multiplicative property
Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later .
